Synchronization of chaotic systems with input constraint

Abstract

It is well known that the dynamics of the chaotic system is very sensitive to the initial conditions of the state, and the synchronization of two identical chaotic systems is only obtained, in general, with the high gain control law once their initial conditions are in a certain large deviation. Furthermore, the initial conditions are commonly unknown in practice, which causes difficulty in synchronizing two chaotic systems. This paper deals with the synchronization of two unified chaotic systems with input constraint. First, the scalar sign function is utilized to approximate the constrained non-smooth input function so that a continuous smooth nonlinear input function and an approximated nonlinear synchronized error system are obtained. Then, an optimal linear quadratic regulator (LQR) continuous-time control law is designed based on the optimal linear model, which is constructed at the sampled operating point of the afore-mentioned approximated nonlinear synchronized error system. To reduce the high magnitude of the obtained control law, the continuous-time control law is digitally redesigned for the implementation and an iterative procedure is proposed to adjust the weighting matrices in the LQR performance index so as to avoid input saturation occurs. Finally, three illustrative examples of the Lorenz, the Chen and the L chaotic systems decomposed from the unified chaotic system are given to demonstrate the effectiveness of the proposed method.